普林斯顿大学博弈论讲义10(5)

普林斯顿大学博弈论讲义3-10

Thus,in the game of Figure1,both(a1a2,A)and(d1d2,D)are Nash equilibria.

Observe that a strategy indicates choices even at histories which previous choices dictated by the same strategy prevent from obtaining.In the game of Figure1,for instance,d1a1is a strategy of Player1,although the history(a1,A)cannot obtain if Player1chooses d1at?.

It stands to reason that d2in the strategy d1d2cannot really be a description of Player 1’s action—she will never really play d2!

We shall return to this point in the next lecture.For the time being,let us provisionally say that d2in the context of the equilibrium(d1d2,D)represents only Player2’s beliefs about Player1’s action in the counterfactual event that she chooses a1at?,and Player2follows it with A.

The key observation here is that this belief is crucial in sustaining(d1d2,D)as a Nash equilibrium.

Games with observable actions and chance moves

The beauty of the OR notation becomes manifest once one adds the possibility that more than one player might choose an action simultaneously at a given history.The resulting game is no longer one of perfect information,because there is some degree of strategic uncertainty. Yet,we maintain the assumption that histories are observable:that is,every player on the move at a history h observes all previous actions and action pro?les which comprise h.

The OR de?nition is a bit vague,so let me provide a rigorous,inductive one.I also add the possibility of chance moves,i.e.exogenous uncertainty.

De?nition5An extensive-form game with observable actions and chance moves is a tuple Γ=(N,A,H,P,Z,U,f c)where:

N is a set of players;Chance,denoted by c,is regarded as an additional player,so c∈N.

A is a set of actions

H is a set of sequences whose elements are points in i∈J A for some A?N∪{c};

Z and X are as in De?nition1;

P is the player correspondence P:X?N∪{c}

U:Z→R N as in De?nition1;

H satis?es the conditions in De?nition1.Moreover,for every k≥1,(a1,...,a k)∈H implies that(a1,...,a k?1)∈H and a k∈ i∈P((a1,...,a k?1))A.

For every i∈N∪{c},let A i(h)={a i∈A:?a?i∈ j∈P(h)\{i}A s.t.(h,(a i,a?i))∈H}. Then f c:{h:c∈P(h)}→?(A)indicates the probability of each chance move,and f c(h)(A i(h))=1for all h such that c∈P(h).

The de?nition is apparently complicated,but the underlying construction is rather nat-ural:at each stage,we allow more than one player(including Chance)to pick an action;the

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